Abstract:We derive the one-loop correction to the space-time energy of a folded string in AdS 4 × CP 3 carrying spin S in AdS 4 and angular momentum J in CP 3 in the long string approximation. From this general result in the limit J ≪ log S we obtain the one-loop correction to the cusp anomalous dimension which turns out to be − 5 log 2 2π . This value appears to be in conflict with the prediction from the recently conjectured all-loop Bethe ansatz.

We construct zero-curvature representations for the equations of motion of a class of σ-models with complex homogeneous target spaces, not necessarily symmetric. We show that in the symmetric case the proposed flat connection is gauge-equivalent to the conventional one. * Emails: dmitri.bykov@aei.mpg.de, dbykov@mi.ras.ru 1 Generalizations to non-simple groups G are possible, but will be considered elsewhere.

In the present paper we revisit the so-called Haldane limit, i.e. a particular continuum limit, which leads from a spin chain to a sigma model. We use the coherent state formulation of the path integral to reduce the problem to a semiclassical one, which leads us to the observation that the Haldane limit is closely related to a Lagrangian embedding into the classical phase space of the spin chain. Using this property, we find a spin chain whose limit produces a relativistic sigma model with target space the manifold of complete flags U(N)/U(1) N . We discuss possible other future applications of Lagrangian/isotropic embeddings in this context.